Intersection numbers of Hecke cycles on Hilbert modular varieties

INTERSECTION NUMBERS OF HECKE CYCLES ON HILBERT MODULAR VARIETIES By JAYCE GETZ Abstract. Let O be the ring of integers of a totally real number field E and set G := ResE/Q( GL2). Fix an ideal c ⊂ O. For each ideal m ⊂ O let T(m) denote the mth Hecke operator associated to the standard compact open subgroup U0(c) of G(Af ). Setting X0(c) := G(Q)\G(A)/K∞U0(c), where K∞ is a certain subgroup of G(R), we use T(m) to define a Hecke cycle Z(m) ∈ IH2[E:Q](X0(c) × X0(c)). Here IH• denotes intersection homology. We use Zucker’s conjecture (proven by Looijenga and independently by Saper and Stern) to obtain a formula relating the intersection number Z(m) · Z(n) to the trace of ∗T(m) ◦ T(n) considered as an endomorphism of the space of Hilbert cusp forms on U0(c). 1. Introduction. We begin by recalling the main theorem of Hirzebruch and Zagier’s famous paper [HZ]. In their paper they examine the intersection numbers of certain “Hirzebruch-Zagier cycles” Zm. These cycles are sums of the closures of (affine) modular curves and certain compact Shimura curves inside a toroidal compactification of the Hilbert modular surface SL2 (OQ( √ p))\H2. Here H is the usual complex upper half plane, p ≡ 1 (mod 4) is a prime and OQ( √ p) is the ring of integers of Q( √ p). In particular, if Zm · Zn denotes the “number of intersections” of Zm and Zn, then the bulk of [HZ] is devoted to proving that for each m ∈ Z>0 the generating series ∞
n=0 (Zm · Zn)qn (1.1) is a weight 2 modular form for Γ0(p) with character (p· ). Here Zm·Z0 is essentially the volume of Zm and q := e2πiz for z ∈ H. Manuscript received October 16, 2005; revised August 2, 2006. Research supported in part by the ARO through the NDSEG Fellowship program. American Journal of Mathematics 129 (2007), 1623–1658. c 2007 by The Johns Hopkins University Press. 1623 1624 JAYCE GETZ In this paper we provide a kind of generalization of Hirzebruch-Zagier. Since their seminal work, a number of cohomology theories useful for studying topological intersection theory on Hermitian locally symmetric spaces such as Hilbert modular varieties have developed, including intersection homology, L2-cohomology, and Harder’s theory of Eisenstein cohomology. These theories have intrinsic interest, but our treatment of them is utilitarian; they provide a natural framework for studying the interplay between intersection theory on modular varieties and the coefficients of modular forms. Our goal is to revisit HirzebruchZagier armed with these theories with the idea of proving comparison theorems for pairings that arise naturally in the context of intersection homology on the one hand, and Hecke algebras on the other. Concretely, we consider the graphs of Hecke correspondences on the product of two Hilbert modular varieties associated to totally real fields of arbitrary dimension. In order to make this precise, we must develop some notation: Let E be a totally real number field with degree [E : Q] = n, ring of integers O, and narrow class number h+; thus h+ is the order of the ray class group modulo β∞, the product of the infinite places. To every ideal c ⊂ O we associate in (2.3) the Hilbert modular variety Y0(c) := h+  j=1 Γj\Hn via a well-known construction (see [Ge, §I.7]). This variety has h+ components, each of complex dimension n (the Γj are discrete subgroups of ( GL2 (R)+)n). For convenience, denote its Baily-Borel compactification by X0(c). We then define in§ 5, for every ideal m ⊂ O, a Hecke cycle Z(m) ∈ IH2n(X0(c) × X0(c)). Here IH• denotes intersection homology (with middle perversity, see § 4). The class Z(m) is determined by the Hecke operator T(m) for X0(c) (see (3.6) for the definition of T(m)). In (2.2) we recall the definition of the standard compact open subgroups U0(c) ≤ (ResE/Q GL2)(Af ), where c ⊂ O is an ideal as above and Af denotes the finite adèles of Q. Let Tc denote the Hecke algebra associated...